| Sooch | | apprenti | | 1 message posté |
|  Posté le 08-03-2007 à 21:36:04    
| How about a little look into the past.
(Ben will translate this post into French soon).
Let us turn this circle around and begin a discussion of the beginnings of PI (π).
The fact that the ratio of the circumference of a circle to it’s diameter is constant has been known for such a long time, that it is quite hard to trace conclusively the true origins.
The earliest values of PI including the 'Biblical' value of 3, were almost certainly produced by measurement. In a list of specifications for the great temple of Solomon, built around 950 BC it was given that PI = 3. Not a very accurate value of course and not even very accurate in its day, in that the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates.
In the Egyptian Rhind Papyrus which is dated about 1650 BC, there is good evidence for 4 x (8/9)^2 = 3.16 as a value for π (PI).
The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). Archimedes was the greatest mathematician of his age. His contributions in geometry revolutionized the subject and his methods anticipated the integral calculus 2,000 years before Newton and Leibniz. He was also a thoroughly practical man who invented a wide variety of machines including pulleys and the Archimidean screw pumping device.
He obtained the approximation 223/71 < π < 22/7.
Before giving an indication of his proof, notice the remarkable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds, we obtain 3.1418, an error of about 0.0002.
Here is Archimedes’ argument.
Consider a circle of radius 1, in which we inscribe a regular polygon of 3 x 2^n-1 sides, with semiperimeter b , and superscribe a regular polygon of 3 x 2^n-1 sides, with semiperimeter a .
The diagram for the case n = 2 will be posted soon.
The effect of this procedure is to define an increasing sequence
b1 , b2 , b3 , ...
and a decreasing sequence
a1 , a2 , a3 , ...
such that both sequences have limit π.
Using trigonometrical notation, we see that the two semi-perimeters are given by
a[n] = K tan(π/K), b[n] = K sin(π/K), where K = 3 x 2^n-1.
Equally, we have a[n]+1 = 2K tan(π/2K), b[n]+1 = 2K sin(π/2K),
and it is not a difficult exercise in trigonometry to show that
(1/a[n] + 1/b[n]) = 2/an+1 . . . (1)
a[n]+1b[n] = (b[n]+1)2 . . . (2)
Archimedes starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that
b[6] < π < a[6] .
It is important to realize that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a truly remarkable feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.
Message édité le 13-03-2007 à 21:42:18 par Sooch |
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